Lipschitz rigidity for scalar curvature

Lipschitz rigidity for scalar curvature

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Bernhard Hanke, University of Augsburg and IAS
Fine Hall 314

Lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable rigidity properties determined by spectral properties of the Dirac operator. For instance, a fundamental result of Llarull states that there is no smooth Riemannian metric on the n-sphere which dominates the round metric and whose scalar curvature is greater than or equal to the scalar curvature of the round metric, except the round metric itself. A similar result holds for smooth comparison maps from spin Riemannian manifolds to round spheres.  

In a joint work with Simone Cecchini and Thomas Schick, we generalize this result to Riemannian metrics with regularity less than C^1 and Lipschitz comparison maps, answering a question of Gromov in his "Four Lectures". To this end, we rely on a notion of scalar curvature in the distributional sense introduced by Lee-LeFloch and on spectral properties of Lipschitz Dirac operators.  It turns out that the existence of a nonzero harmonic spinor field - guaranteed by the Atiyah-Singer index theorem - forces the given comparison map to be quasiregular in the sense of Reshetnyak.